check parameters

latex/rode_conv_em.tex

@@ -1052,7 +1052,7 @@ \subsection{Homogeneous linear equation with Wiener noise}
     \label{figlinearhomogeneousrode}
 \end{figure}
 
-\cref{tablinearhomogeneousrode} shows the estimated strong error obtained from the Monte Carlo simulations for each chosen time step, with initial condition $X_0 \sim \mathcal{N}(0, 1)$, on the interval $[0, T]$. \cref{figlinearhomogeneousrode} illustrates the order of convergence.
+\cref{tablinearhomogeneousrode} shows the estimated strong error obtained from the $M = 200$ Monte Carlo simulations for each chosen time step $N_i = 2^4, \ldots, 2^{14}$, with initial condition $X_0 \sim \mathcal{N}(0, 1)$, on the time interval $[0, T] = [0.0, 1.0]$. The target resolution is set to $N_{\textrm{tgt}} = 2^{16}$, since we have the exact distribution for it. \cref{figlinearhomogeneousrode} illustrates the order of convergence.
 
 \subsection{Non-homogeneous linear system of RODEs with different types of noises}
 
@@ -1101,9 +1101,9 @@ \subsection{Non-homogeneous linear system of RODEs with different types of noise
     \label{figallnoises}
 \end{figure}
 
-\cref{taballnoises} shows the estimated strong error obtained from the Monte Carlo simulations. \cref{figallnoises} illustrates the order of convergence.
+\cref{taballnoises} shows the estimated strong error obtained from the $M = 80$ Monte Carlo simulations for each chosen time step $N_i = 2^6, \ldots, 2^9$,  on the time interval $[0, T] = [0.0, 1.0]$. All coordinates of the initial condition are normally distributed with mean zero and variance 1. The target resolution is set to $N_{\textrm{tgt}} = 2^{18}$.
 
-Finally, \cref{figsamplepathsallnoises} illustrates sample paths of all the noises used in this system.
+\cref{figallnoises} illustrates the obtained order of convergence, while \cref{figsamplepathsallnoises} illustrates some sample paths of all the noises used in this system.
 
 \begin{figure}[htb]
     \includegraphics[scale=0.6]{img/noisepath_allnoises.png}
@@ -1268,7 +1268,7 @@ \subsection{Fractional Brownian motion noise}
 \end{equation}
 computing a distributionally exact solution of this form is a delicate process. Thus we check the convergence numerically by comparing the approximations with another Euler approximation on a much finer mesh.
 
-More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$,  $N = 64,$ $128,$ $256,$ and $512,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching reasonably well the theoretical estimate of $p = \min\{H+1/2, 1\}.$
+More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, the initial condition as $X_0 \sim \mathcal{N}(0, 1),$ set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$, with $N = 2^6, \ldots, 2^9,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching the theoretical estimate of $p = \min\{H+1/2, 1\}.$
 
 \begin{table}
     \begin{tabular}[htb]{|c|c|c|c|}
@@ -1345,7 +1345,7 @@ \subsection{Population dynamics with harvest}
 
 The function $f=f(t, x, y)$ is continuously differentiable infinitely many times and with bounded derivatives within the positively invariant region. Hence, within the region of interest, all the conditions of \cref{thmmixedcasepractical} hold and the Euler method is of strong order 1.
 
-Below, we simulate numerically the solutions of the above problem, with $\gamma = 0.8,$ $\varepsilon = 0.3,$ $r = 1.0,$ and $\alpha = \gamma r = 0.64$ The geometric Brownian motion process $\{G_t\}_{t\geq 0}$ is taken with drift coefficient $\mu = 1.0,$ diffusion coefficient $\sigma = 0.8,$ and initial condition $y_0 = 1.0.$ The Poisson process $\{N_t\}_{t \geq 0}$ is taken with rate $\lambda = 15.0$. And the step process $\{H_t\}_{t \geq 0}$ is taken with steps following a Beta distribution with shape parameters $\alpha = 5.0$ and $\beta = 7.0$. The initial condition $X_0$ is taken to be a Beta distribution with shape parameters $\alpha = 7.0$ and $\beta = 5.0$, hence we can take $R = 1$. We take $M = 200$ samples for the Monte-Carlo estimate of the strong error of convergence. For the target solution, we solve the equation with a time mesh with $N_{\mathrm{tgt}} = 2^{18}$ points, while for the approximations we take $N = 2^i$, for $i=4, \ldots, 9$.
+Below, we simulate numerically the solutions of the above problem, with $\gamma = 0.8,$ $\varepsilon = 0.3,$ $r = 1.0,$ and $\alpha = \gamma r = 0.64$ The geometric Brownian motion process $\{G_t\}_{t\geq 0}$ is taken with drift coefficient $\mu = 1.0,$ diffusion coefficient $\sigma = 0.8,$ and initial condition $y_0 = 1.0.$ The Poisson process $\{N_t\}_{t \geq 0}$ is taken with rate $\lambda = 15.0$. And the step process $\{H_t\}_{t \geq 0}$ is taken with steps following a Beta distribution with shape parameters $\alpha = 5.0$ and $\beta = 7.0$. The initial condition $X_0$ is taken to be a Beta distribution with shape parameters $\alpha = 7.0$ and $\beta = 5.0$, hence we can take $R = 1$. We take $M = 200$ samples for the Monte-Carlo estimate of the strong error of convergence. For the target solution, we solve the equation with a time mesh with $N_{\mathrm{tgt}} = 2^{18}$, while for the approximations we take $N = 2^i$, for $i=4, \ldots, 9$.
 
 Notice that we can write
 \[
@@ -1490,7 +1490,7 @@ \subsection{A toggle-switch model for gene expression}
 
 We do not have an explicit solution for the equation so we use as target for the convergence an approximate solution via Euler method at a much higher resolution.
 
-For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=4, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 200.$ \cref{tabletoggleswitch} shows the estimated strong error obtained with this setup, while \cref{figtoggleswitch} illustrates the order of convergence. \cref{figtoggleswitchevolution} shows a sample solution, while \cref{figtoggleswitchnoise} illustrates the two components $(A_t, B_t)$ of a sample noise.
+For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=5, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 100.$ \cref{tabletoggleswitch} shows the estimated strong error obtained with this setup, while \cref{figtoggleswitch} illustrates the order of convergence. \cref{figtoggleswitchevolution} shows a sample solution, while \cref{figtoggleswitchnoise} illustrates the two components $(A_t, B_t)$ of a sample noise.
 
 \begin{table}
     \begin{tabular}[htb]{|r|l|l|l|}
@@ -1602,7 +1602,7 @@ \subsection{An actuarial risk model}
 
 For the numerical simulations, we use $O_0 = 0$, $\nu = 5$ and $\varepsilon = 0.8$, for the Ornstein-Uhlenbeck process $\{O_t\}_t$; $\lambda = 8.0$ and $C_i \sim \mathrm{Uniform}(0, 0.2)$, for the compound Poisson process $\{C_t\}$; $R_0 = 0.2$, $\mu = 0.02$ and $\sigma = 0.4$, for the interest rate process $\{R_t\}_t$; and we take $X_0 = 1.0$, so that $U_0 = X_0 + O_0 + R_0 = 1.2$. We set $T = 3.0,$ as the final time.
 
-For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=6, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 100.$ \cref{tableriskmodel} shows the estimated strong error obtained with this setup, while \cref{figriskmodel} illustrates the order of convergence. \cref{figriskmodelnoise} shows a sample path of the noise, which is composed of three processes, while \cref{figriskmodelsurplus} shows a sample path of the surplus.
+For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=6, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 400.$ \cref{tableriskmodel} shows the estimated strong error obtained with this setup, while \cref{figriskmodel} illustrates the order of convergence. \cref{figriskmodelnoise} shows a sample path of the noise, which is composed of three processes, while \cref{figriskmodelsurplus} shows a sample path of the surplus.
 
 \begin{table}
     \begin{tabular}[htb]{|r|l|l|l|}

latex/rode_conv_em_short.tex

@@ -1003,7 +1003,9 @@ \subsection{Non-homogeneous linear system of RODEs with different types of noise
     \label{figallnoises}
 \end{figure}
 
-\cref{taballnoises} shows the estimated strong error obtained from the Monte Carlo simultions. \cref{figallnoises} illustrates the order of convergence and sample paths of all the noises used in this system.
+\cref{taballnoises} shows the estimated strong error obtained from the $M = 80$ Monte Carlo simulations for each chosen time step $N_i = 2^6, \ldots, 2^9$,  on the time interval $[0, T] = [0.0, 1.0]$. All coordinates of the initial condition are normally distributed with mean zero and variance 1. The target resolution is set to $N_{\textrm{tgt}} = 2^{18}$.
+
+\cref{figallnoises} illustrates the order of convergence and sample paths of all the noises used in this system.
 
 The strong order 1 convergence is not a suprise in the case of the Wiener and Ornstein-Uhlenbeck process since the corresponding RODE can be turned into an SDE with an additive noise. In this case, the Euler-Maruyama approximation for the noise part of the SDE is distributionally exact, and the Euler method for the RODE becomes equivalent to the Euler-Maruyama method for the SDE, and it is known that the Euler-Maruyama method for an SDE with additive noise is of strong order 1 \cite{HighamKloeden2021}. For the remaining noises, however, previous works would estimate the order of convergence to be below the order 1 attained here.
 
@@ -1050,7 +1052,7 @@ \subsection{Fractional Brownian motion noise}
 \end{equation}
 computing a distributionally exact solution of this form is a delicate process. Thus we check the convergence numerically by comparing the approximations with another Euler approximation on a much finer mesh.
 
-More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$,  $N = 64,$ $128,$ $256,$ and $512,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching reasonably well the theoretical estimate of $p = \min\{H+1/2, 1\}.$
+More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, the initial condition as $X_0 \sim \mathcal{N}(0, 1),$ set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$, with $N = 2^6, \ldots, 2^9,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching the theoretical estimate of $p = \min\{H+1/2, 1\}.$
 
 \begin{table}
     \begin{tabular}[htb]{|c|c|c|c|}